What is the root of the polynomial equation x(x - 2)(x + 3)=18 ? Use a graphing calculator and a system of equations.

To find the roots of the polynomial equation \(x(x - 2)(x + 3) = 18\), we'll use a step-by-step approach. Here's how to solve it:

### Step 1: Expand the polynomial
First, expand the left-hand side of the equation to make it easier to work with:

\[
x(x - 2)(x + 3) = x[(x^2 + 3x) - 2(x + 3)] = x(x^2 + 3x - 2x - 6) = x(x^2 + x - 6)
\]

So the equation becomes:

\[
x^3 + x^2 - 6x = 18
\]

### Step 2: Set the equation equal to zero
Move all terms to one side of the equation:

\[
x^3 + x^2 - 6x - 18 = 0
\]

This is a cubic equation.

### Step 3: Graphical approach
Use a graphing calculator to graph the function \(f(x) = x^3 + x^2 - 6x - 18\). The points where the graph intersects the x-axis will give the roots of the equation.

Upon graphing, you would find the approximate roots.

### Step 4: Use a system of equations
To verify the roots, let's use a system of equations to check one of the potential solutions. Since graphing provides approximate values for \(x\), you can substitute these values back into the equation to ensure that they satisfy the original equation.

#### Example system for solving roots:
Let \( x = r \) be one of the roots from the graphing calculator. Substitute \(r\) into the polynomial equation to see if it holds:

\[
r^3 + r^2 - 6r - 18 = 0
\]

If this equation is true for your chosen \(r\), that value is indeed a root.

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Would you like to go into more detail about solving this equation graphically or algebraically? Or check a specific root?

Here are some questions to deepen your understanding:

1. How do you interpret the roots of a cubic equation graphically?
2. What are the differences between algebraic and graphical methods for solving equations?
3. How can you apply the Rational Root Theorem to help find the roots of this cubic equation?
4. What other methods can be used to find the exact roots of a cubic equation?
5. How would this equation change if the constant on the right-hand side were different, say 0 or 36?

**Tip**: When using a graphing calculator to solve polynomial equations, zooming in on the graph can help identify the roots more precisely, especially when dealing with approximate values.

What is the root of the polynomial equation x(x - 2)(x + 3) = 18? Use a graphing calculator and a system of equations.

Learn how to solve the cubic polynomial equation x(x - 2)(x + 3) = 18 using both a graphing calculator and a system of equations. This guide covers polynomial expansion, graphing methods, and system-solving techniques, suitable for high school students in Grades 10-12.

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